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Gradient blowup rate for a semilinear parabolic equation. (English) Zbl 1191.35074
The paper is devoted to the initial boundary value problem for the heat equation with a nonlinear gradient source term: $u_t = u_{xx} + x^m|u_x|^p$, $0<x<1$, $t>0$, for which the spatial derivative of solutions becomes unbounded in finite time while the solutions remain bounded. It is shown that the spatial derivative of solutions is globally bounded in the case $p\le m+2$ while blowup occurs at the boundary when $p > m+2$. The authors prove that, with an additional restriction on $p$ and the assumption on the initial data so that the solution is monotonically increasing both in time and in space, both the lower and upper bound of the blowup rate are $(T-t)^{-(m+1)/(p-m-2)}$.

35B44Blow-up (PDE)
35K58Semilinear parabolic equations
35K20Second order parabolic equations, initial boundary value problems
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